Four-current

The four-current is the Lorentz four-vector J^μ = (cρ, J) that combines electric charge density ρ (the time component, multiplied by c for unit-matching) and current density J (the spatial three-components) into a single covariant object. Under a Lorentz boost, ρ and J mix exactly the way the components of any four-vector (ct, x) mix — a fact that turns the apparently separate concepts of "stationary charge" and "flowing current" into manifestations of the same underlying object viewed from different frames. A static charge distribution in the lab frame becomes both a charge density and a current density in any boosted frame; conversely, a pure current in one frame may not be electrically neutral in another.

The four-current sources the electromagnetic field tensor F^{μν} via the inhomogeneous Maxwell equation in covariant form: ∂_μ F^{μν} = μ₀ J^ν. With ν = 0 this reproduces Gauss's law ∇·E = ρ/ε₀; with ν = i ∈ {1, 2, 3} it reproduces the Ampère–Maxwell law ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. The continuity equation ∂_μ J^μ = 0, which expresses local charge conservation, follows automatically from the antisymmetry of F^{μν}: contracting both indices of ∂_μ ∂_ν F^{μν} kills the right-hand side because partials commute and F^{μν} is antisymmetric. Charge conservation is therefore not an extra postulate but a structural consequence of the Lagrangian formulation — Noether's theorem applied to the gauge symmetry A_μ → A_μ + ∂_μΛ.